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Mathematics 2017, 5(4), 70; https://doi.org/10.3390/math5040070

Controlling Chaos—Forced van der Pol Equation

1
Air Force Research Laboratory, Albuquerque, NM 87117, USA
2
Air Force Research Laboratory, Dayton, OH 45433, USA
3
Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA
*
Author to whom correspondence should be addressed.
Received: 6 October 2017 / Revised: 14 November 2017 / Accepted: 17 November 2017 / Published: 24 November 2017
(This article belongs to the Special Issue Mathematics on Automation Control Systems)
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Abstract

Nonlinear systems are typically linearized to permit linear feedback control design, but, in some systems, the nonlinearities are so strong that their performance is called chaotic, and linear control designs can be rendered ineffective. One famous example is the van der Pol equation of oscillatory circuits. This study investigates the control design for the forced van der Pol equation using simulations of various control designs for iterated initial conditions. The results of the study highlight that even optimal linear, time-invariant (LTI) control is unable to control the nonlinear van der Pol equation, but idealized nonlinear feedforward control performs quite well after an initial transient effect of the initial conditions. Perhaps the greatest strength of ideal nonlinear control is shown to be the simplicity of analysis. Merely equate coefficients order-of-differentiation insures trajectory tracking in steady-state (following dissipation of transient effects of initial conditions), meanwhile the solution of the time-invariant linear-quadratic optimal control problem with infinite time horizon is needed to reveal constant control gains for a linear-quadratic regulator. Since analytical development is so easy for ideal nonlinear control, this article focuses on numerical demonstrations of trajectory tracking error. View Full-Text
Keywords: van der Pol; oscillator; forced; linear feedback; nonlinear feedforward van der Pol; oscillator; forced; linear feedback; nonlinear feedforward
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Cooper, M.; Heidlauf, P.; Sands, T. Controlling Chaos—Forced van der Pol Equation. Mathematics 2017, 5, 70.

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